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arxiv: 2511.11537 · v2 · pith:ZUDYQOT7new · submitted 2025-11-14 · ✦ hep-ph · hep-th

Two-loop all-plus helicity amplitudes for self-dual Higgs boson with gluons via unitarity cut constraints

Simon Badger , Christian Biello , Colomba Brancaccio , Federico Ripani This is my paper

Pith reviewed 2026-05-17 21:56 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords two-loop amplitudesself-dual Higgsall-plus helicityunitarity cutsheavy top limitpolylogarithmsfinite fieldsgluon amplitudes
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0 comments X

The pith

Two-loop amplitudes for self-dual Higgs with positive helicity gluons are computed via unitarity cut constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the two-loop amplitudes for a self-dual Higgs boson with up to four positive helicity gluons in the heavy top-quark limit. The vanishing of tree amplitudes in the all-plus sector allows the polylogarithmic parts to be built from four-dimensional unitarity cuts to rational one-loop and tree amplitudes. The rational ambiguities are then fixed by tensor integral reduction over finite fields. The results are given in terms of polylogarithms of weight at most two and compact rational functions of spinor-helicity products. These explicit forms make the amplitudes available for use in phenomenological studies.

Core claim

We present the two-loop amplitudes for a self-dual Higgs boson with up to four positive helicity gluons in the heavy top-quark limit. Because the tree amplitudes in the all-plus sector vanish, we can construct simple representations of the polylogarithmic parts of the two-loop amplitudes using four-dimensional unitarity cuts into rational one-loop and tree amplitudes. The remaining rational function ambiguity is extracted from a tensor integral reduction over finite fields. The final expressions are presented using polylogarithms up to weight two and compact rational functions of spinor-helicity products.

What carries the argument

Four-dimensional unitarity cuts into rational one-loop and tree amplitudes, with tensor integral reduction over finite fields to resolve rational ambiguities.

If this is right

  • The polylogarithmic parts of the two-loop amplitudes have simple representations.
  • The remaining rational functions are compact and expressed in spinor-helicity products.
  • Explicit results are provided for amplitudes with one to four gluons.
  • Polylogarithms up to weight two are sufficient for these expressions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These methods could be applied to compute similar amplitudes in other vanishing sectors of scattering processes.
  • The results may help in building complete two-loop predictions for Higgs boson production in association with jets.
  • Independent checks using different computational techniques would strengthen confidence in the expressions.

Load-bearing premise

Tree amplitudes in the all-plus sector vanish.

What would settle it

A calculation of one of these amplitudes using traditional methods that yields a different functional form or numerical value at a test point.

read the original abstract

We present the two-loop amplitudes for a self-dual Higgs boson with up to four positive helicity gluons in the heavy top-quark limit. Because the tree amplitudes in the all-plus sector vanish, we can construct simple representations of the polylogarithmic parts of the two-loop amplitudes using four-dimensional unitarity cuts into rational one-loop and tree amplitudes. The remaining rational function ambiguity is extracted from a tensor integral reduction over finite fields. The final expressions are presented using polylogarithms up to weight two and compact rational functions of spinor-helicity products.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper computes two-loop amplitudes for a self-dual Higgs boson coupled to up to four positive-helicity gluons in the heavy-top limit. It exploits the vanishing of all-plus tree amplitudes to obtain the polylogarithmic content directly from four-dimensional unitarity cuts into one-loop and tree amplitudes, with the remaining rational terms fixed by finite-field tensor reduction. Final expressions use weight-two polylogarithms and compact rational functions of spinor-helicity products.

Significance. If the results hold, they supply new analytic two-loop expressions relevant for precision Higgs phenomenology in the effective theory. The approach demonstrates a clean application of unitarity methods in the all-plus sector without free parameters or post-hoc fitting, and the explicit polylogarithmic-plus-rational form aids reproducibility and further use in cross-section calculations.

major comments (1)
  1. §3.2, around the four-gluon cut: the claim that the polylogarithmic part is fully determined by the 4D cut into the one-loop all-plus amplitude requires an explicit verification that the cut integrand reproduces the known one-loop result before integration; without this check the extraction of the weight-two terms remains unconfirmed for multiplicity four.
minor comments (2)
  1. The finite-field reduction in §4 should specify the prime modulus and the number of independent points used to solve for the rational coefficients, to allow independent reproduction.
  2. Notation for the self-dual Higgs effective vertex (Eq. (2.3)) could be cross-referenced when it first appears in the amplitude expressions to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive comment on the four-gluon cut. We address the point below.

read point-by-point responses

  1. Referee: §3.2, around the four-gluon cut: the claim that the polylogarithmic part is fully determined by the 4D cut into the one-loop all-plus amplitude requires an explicit verification that the cut integrand reproduces the known one-loop result before integration; without this check the extraction of the weight-two terms remains unconfirmed for multiplicity four.

    Authors: We agree that an explicit verification strengthens the presentation. Although the one-loop all-plus four-gluon amplitude is standard in the literature and our four-dimensional cut follows the same unitarity construction used for lower multiplicities, we will add a direct check in the revised manuscript. Specifically, we will show that the integrand obtained from the 4D cut into the known one-loop all-plus amplitude reproduces the expected result prior to integration, thereby confirming the weight-two polylogarithmic content for the four-gluon case. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on the known vanishing of all-plus tree amplitudes to construct polylogarithmic parts directly from four-dimensional unitarity cuts into rational one-loop and tree amplitudes, with rational ambiguities fixed via finite-field tensor integral reduction. These steps invoke standard external techniques (unitarity and integral reduction) rather than any self-defined quantities, fitted parameters from the target result, or load-bearing self-citations. The final expressions are therefore independent of the paper's own inputs and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of perturbative quantum field theory in the effective Higgs-gluon theory, dimensional regularization, and the validity of four-dimensional unitarity cuts for extracting polylogarithmic parts. No new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Four-dimensional unitarity cuts suffice to determine the polylogarithmic content of the two-loop amplitude when tree-level all-plus amplitudes vanish.
    Invoked in the abstract to justify the construction method.
  • domain assumption The heavy top-quark limit is a valid approximation for the self-dual Higgs effective theory.
    Stated as the regime in which the amplitudes are computed.

pith-pipeline@v0.9.0 · 5400 in / 1393 out tokens · 27556 ms · 2026-05-17T21:56:58.501115+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Because the tree amplitudes in the all-plus sector vanish, we can construct simple representations of the polylogarithmic parts of the two-loop amplitudes using four-dimensional unitarity cuts into rational one-loop and tree amplitudes.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit

    hep-ph 2026-05 unverdicted novelty 7.0

    Analytic expressions for the finite remainders of two-loop leading-color helicity amplitudes in Higgs plus two-jet production are obtained in the heavy-top effective theory using numerical unitarity and a new partial-...

  2. Pseudo-Evanescent Feynman Integrals from Local Subtraction

    hep-th 2026-05 conditional novelty 7.0

    Local subtraction reduces pseudo-evanescent Feynman integrals to products of one-loop integrals or one-fold integrals, with the finite part of the two-loop all-plus five-point amplitude arising solely from ultraviolet...

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